Chapter 1.8, Problem 17E

Show that each of these statements can be used to express the fact that there is a unique elementxsuch thatP(x) is true. [Note that we can also write statement as $\exists !xP\left(x\right)$ .]

a) $\exists x\forall y\left(P\left(y\right)\leftrightarrow x=y\right)$

b)

   $\exists xP\left(x\right)\wedge \forall x\forall y\left(P\left(x\right)\wedge P\left(y\right)\to x=y\right)$

c) $\exists x\left(P\left(x\right)\wedge \forall y\left(P\left(y\right)\to x=y\right)\right)$